Can I stay in Thailand on a non-immigrant B visa until April if my work permit expires in February?

You are on an extension of stay which means you signed a form saying you would notify immigration if your circumstances change. What is relevant is the date you will stop working, that is the date you need to cancel your work permit, inform immigration and leave the country. You can normally do this in advance with a letter from your employer then you will need to leave the country on that date. You can then return either visa exempt or with a tourist visa

Guessing you don't have a VISA, you have an extension of stay (issued at immigration inside Thailand), therefore, the day your work permit expires, so does your extension of stay. You will therefore need to leave the country that day, or apply for a new extension based on marriage or retirement for example. Also, you need to cancel both your work permit and extension, this will stop you getting a large fine when you apply again.