Congratulations on graduating with your BSN! As a public health nurse who has been in the field for *cough* years, I can tell you confidently that you could go your whole career in public health without needing to get a master's degree depending on the type of work you want to do. The majority of my peers have their BSN and nothing more for staff level positions, although pretty much every supervisor or program manager has a master's degree whether MPH or MSN, and a few have DNPs. Before embarking on graduate school (and graduate school loans), consider dipping your toe in the field before plunking down for a graduate degree. Disregard that advice if you are independently wealthy, of course.
But as to your original question, I am working full time (40 hours/week), volunteer several times per month for the Red Cross, and carry a full time course load for an MPH program. The first year of my program I also worked a PRN job with an additional 6-8 hours/week on top of my regular position. I don't have children though, so that probably makes it a little easier for me to do. It can certainly be done if you are disciplined. I'd wager that most of the people in my program work full time. You know your limitations best.
If you have your heart set on getting an MPH degree, I would encourage you to explore programs that are accredited. Just like nursing schools have NLN or CCNE accreditation, public health programs are also accredited and are more respectable when they carry that designation. Some jobs in public health, particularly federal ones (CDC, NIH, IHS, etc) will require your degree to be accredited for it to count. You can find a list of accredited programs here: Accredited Schools & Programs | Council on Education for Public Health
There are accredited online programs too (like mine) that may be more flexible for your needs as an adult learner.
Best of luck whatever you decide!