In my opinion, your H is crossing lines with this woman.
As I said upthread, I would not be comfortable with the messages. They seem to be doing a dance with each other.
He is her boss, but is being quite familiar. He is eager to have alone-time with her with the ”early, so we can hang out, have lunch, have dinner, etc.” and the suggestion that they ”stay an extra night and make a day of it.” He assumes that she will be copacetic with all the cozy 1 on 1.
At first she holds him at arm’s length. She mentions her need for a break, probably to not seem too personally interested. At his suggestion to stay an extra night, she says, ”Let’s see.” In the end, though, she says, ”Ok great” to his suggestion of arriving early Sunday for their day/evening togetherness. She could go either way.
I reread what you’ve reported about their history. He thought she was special and pushed for her to be hired. Now he is texting her personal messages, asking if she is okay. She is confiding in him about her issues at home. This is inappropriate and too familiar, and can lead to emotional/physical affair territory.
All in all, I think he is over-invested emotionally and is fishing for more. If you come to the same conclusion, how will you handle this situation?