Question is more or less in the title. I've googled and googled to no avail. Unfortunately I can not try it out myself for a few more days.
Yes it can. I can make a request using HttpClient to an endpoint with a CookieContainer attached to my HttpClientHandler and when I get the response back I can look in the HttpClientHandler instance's CookieContainer and the cookie is there.
I have had god awful trouble with cookies but it usually ends up being an issue with using the correct domain. Everything has been working for me now for a while, so it is definitely possible.