Can I still work legally if I marry my boyfriend who has a green card?

You can continue to work in H-1B status if you get married. That is not a problem but you do not get work authorization through the marriage. If you get married before your boy friend gets his green card, you can get one with him. You must maintain your own status if he is already a permanent resident but his green card does not prevent you from pursuing one on your own through your employment.

Most H-1B's will qualify for permanent residence under the EB third preference. It takes about 6 months for the labor certification and then there is a 6 year backlog (for most countries) for the priority date to be current to allow for the application for permanent residence. If you marry an LPR, it generally takes about three years for the priority date to be current to file for permanent residence. Please note that if you apply through your LPR spouse, you must maintain your nonimmigrant visa status (for example H-1B) until the FB priority date is current and you actually file.