That’s an interesting question, my dear guest and I’m really glad you asked. Well, when it comes to demons I’m rather talkative (*/ω＼*)
Warning: contains spoilers of everything because I’m basing on different sources!
First of all, we need to know how does the demons’ nature actually work. Unfortunately, the Black Butler itself isn’t showing us the background of their lifes nor even their personality. It’s pretty visible in the Sebastian’s character; all we know about him it that, he is always playing, trying to act as a butler in the best way he can and basically everything he’s doing goes for Ciel’s liking.
There was a headcanon somewhere that he’s probably a terrible cook since every meal he’s made is for the young Phantomhive and people have different tastes when it comes to eating. That’s why I’d rather base on mythology where the demons were mentioned.
The demons are usually considered as malevolent beings, wishing and making the worst they can, causing pain and suffering. However, if you’d do some research in the folklore and myths of different ethnic goups from the world, you’d be surprised that not all demons were as bad as they’re described. Of course, humans were always scared of them, terrified by their power which could create as well as destroy. That’s why in some places they were worshipped so they wouldn’t have a reason to do any harm and so even become protective.
Let me introduce you my favourite demon from Slavic mythology - Domownik (Domowoj). Even though he was called a demon, his presence in the houses was welcomed and all what the residents had to do was to treat him as a part of family.
It could be hard since he was more audiable that visible tho. Keept in the good mood he was blessing the house, leaving peace, looking after children and helping in the chores. However, when ignored he could take a revenge on the household by crushing plates, scaring and haunting, simply acting like poltergeist. Still, it doesn’t change the fact he just needed some attention to be a house guardian. Fun fact is that, even today on the Christmas Eve there’s a custom in Slavic houses to keep one extra cover for the ‘unexpected guest’ but not everybody knows that originally it was supposed to be for the Domownik, being an invisible part of the family.
Therefore, it’s clear that demons weren’t simply black and white, evil to the bones,
more likely they were just demanding and capricious. Just like humans
are! It can prove that they were able to feel emotions, moreover similiar to ours.
The next source, which was kind of confirmed by the Black Butler Live Action (HERE) is that, the demons are in truth fallen angels. Long story short, pure being commits a sin, falls from heaven and then becomes a corrupted creature. The question is, don’t they remember their previous way of life? Or don’t they want to remember? The love itself is always considered as positive emotion, perfectly suitable for merciful, innocent angels so the inquire about what happens with that after the fall is obvious. Is it taken from them or stays somewhere deep hidden in their dead hearts? There I’m going to refer to the secon season of Black Butler anime because Hannah Anafeloz may give some clue about that. She, being the demon was crying after eating Lucas’ soul, it’s clearly visible that this boy surprised and moved/touched her causing to actually feel this kind of emotions. It’s sort of colliding with Sebastian’s words in chapter 89. Even though he said: “I’m not even capable of feeling emotions that would cause me to shed these [tears]…” it doesn’t change the fact, Hannah actually could do it and was capable of having feelings like those. Basically, it’s not a feature of every demon to be always unmoved,
probably it’s just that Sebastian is an asshole. And finally, we got to the point, what about Claude?
According to what I wrote above, it’s not impossible for Claude to feel some kind of good emotions, however, I think it highly depends on his personality. He was presented as cruel being, perfect simple for the demons’ definition - malevolent and greedy. Let’s not forget that he was probably really hungry also, which could affect on his behaviour negatively.
We all can be mad when there’s nothing good to eat, right? Unlike Sebastian, he seems to like his nature and accept it, often acting like a true demon he is, while the Phantomhive butler enjoys compliments, uses his supernatural powers to impress humans and just to be the best in everything. Claude, on the other hand, isn’t playing, he is lying. Personally, I think the hardest thing is to catch his eye and don’t bore him later. The said human in who he could eventually fall in love with must be surprising to him: have different taste, opinions, behaviour, act in a way he didn’t see before AND he likes. I wouldn’t say it’s completely impossible for him to love because of his nature since being the demon isn’t excluding that, in my honest opinion. If the theory about being fallen angel is true also, it gives even more chances to make it true by making his fallen heart burn again.
And the last but not least, Claude seems to be bored. He’s bored with his life which is clearly visible by his always impassive expression, he’s bored with his master and he doesn’t show him too much affection, he’s searching for something new which in that case happened to be Ciel’s soul, he’s longing for new taste, he’s smiling while fighting with Sebastian on an island because finally there’s something which is giving him shivers of emotions. It’s not really that he’s plainly evil, he’s just looking for something to excite him which is hard because of the amount of time he’s living and walking on Earth. Being immortal and invincible is his worst curse and he can’t find a way out of this. If there’d be a human who could entertain him, show him something new or at least speak in a different, interesting way, he probably could fall in love with them.
Furthermore, the love could be such an interesting adventure for him.