The answer is actually quite obvious:
If you start the conversation (after the hello/how are you?/fine you?/ok ok ...) with a genuine interest in how they feel, you'll achieve exactly what you are asking for here. Asking how they would feel about you talking happily about your work shows them 2 things:
1) That you are willing to give them attention (see 'Chillin's answer, they covered that part really well already).
2) That you are trying to be considerate but don't know the best way.
I don't have experience with talking to homeless people myself but I do have experience talking with disabled or otherwise unfortunate people. It has always turned out great when I asked them how they feel when talking about the "normal" days.
Sometimes they don't want to be reminded so you can look together for other subjects you both feel comfortable talking about. In case of my lonely handicapped friend, she actually loved hearing about the positive things happening in my life. By talking in the same way as I would with anyone else, she get's to experience being "normal" herself. It made her feel that I truly treated her as a normal person, so that she was more than her disability.
We can't know for sure how your homeless acquaintance would like talking about your work, but I wouldn't be surprised if he would love it, even if it's just so he can feel normal during your talk(s) again.
The only ones that can answer that question, how they feel talking about certain topics, are they themselves.
OP asked "How to do the asking part" when you're shy and would feel weird directly asking.
I don't really know how to overcome your own shyness. My own solution is just do it and if it's taken badly explain afterwards that you didn't intend it that way but are genuinely curious.
You could go with something like this:
[saying hi/how are you /...]
I've been meaning to ask you something ... I'd like to talk with you about my everyday life but I'm a bit afraid of how you'd take it. Would you enjoy hearing about the happy stuff that happened to me or would you prefer talking about other things?
Give them some time afterwards if needed. Preferably put a gentle smile on your face to show that you really are interested in talking with them. Then they can respond however way they want.
Based on absolutely no experience with homeless people but on my experience with friends/family who are locked up in their house and are deprived of human contact outside of a random monthly visit I would expect one of the folowing responses:
In case they've been deprived of basic human interactions for way longer than anyone would dare to admit (hopefuly really unlikely):
- [start crying] ... thank you ... (<- good time to offer a hug if you are ok with that too).
In case they're just mostly shutting themselves out from reality yet don't want any trouble:
- Oh don't bother, I don't mind being on my own, you're better of chatting with your friends instead.
- [other generic response that tries to get you to move along without starting a discussion]
What I personally think is the most likely response:
- Oh I'd love to hear about you more! Please do tell.
You really can't do much wrong when you're showing genuine interest in people. There may be some uncomfortable moments yes, but overall it'll be a positive experience. You'll just have to try and then go from there.