You become HSA-eligible as of the 1st day of the 1st month that you meet all HSA-eligibility criteria. So if you have a qualifying HDHP that begins on March 15, you don’t become HSA-eligible until April 1.
Alternatively, if you lose HDHP coverage, you maintain HSA-eligibility until the end of the month in which you lose such coverage.
Source: IRC Sec. 223(c)(1)(A)(i)