My sense is that Hedda has no concept of what "home" really is. We get the sense that George and his aunt are family, as well as Berta. The relationship between these people has been established over time, based upon a mutual love for each other. No matter what building these people would live in, they would always be at home with each other because of their concern and admiration for each other.
Hedda obviously does not fit in. It is not because she has not been welcomed or because she is shy. Hedda shows no sign of wanting to be a part of the family: in fact, she seems to resent it and begins by trying to break things up making Aunt Julia feel unwelcome and insisting that Berta be fired. Hedda does not understand the compassion people feel for others, and this may well indicate that a family connection is something she neither understands or wants. She only wants things her way.
Hedda would never feel at home in George's house or with his family for she seems incapable of feeling love for others. George's home is a haven, a place of acceptance and support, and Hedda does not fit in there...by choice.
The concept of home is not a physical building, but an emotional place. As I noted, George, Julia and Berta would be at home in any building they lived in.