"I do not have a boyfriend."
I'm not 100% comfortable with this explanation. Do you have any external sources? This "use は with negatives" thing seems to be thrown around a lot as a "rule of thumb" that I doubt actually reflects how は and が are practically used. (Thanks, of course, for attempting to explain; I'm just wondering if you have any other sources that might go into more depth.)
Learn Japanese Adventure talks about changing the particle to は in negative sentences in detail.
Let's see how to answer this Yes/No question as follow...
Rule 1: None/を/が/も → は
Meaning: When there is no particle or the particle is either を, が or も, change it to は.
Rule 2: Other Particles → Particle + は
Meaning: If the particle is neither を, が nor も, add は to the particle. (に → には, と → とは, で → では, へ → へは, etc)
Please take note that the above rules apply only if you are answering a Yes/No question in negative ways. Normal negative sentence would not need to change particle or add は (wa) to other Japanese particles.
However, sometimes even for normal negative sentence, people change the Japanese particle in order to emphasize something.
One of the native Japanese speakers on the forums posted this interesting Japanese article from NHK where a survey found that 84% of the participants expected the negative form of the verb to follow after the phrase きょうは、天気は, and 56% expected the positive form of the verb to follow after the phrase きょうは、天気が.
[Edit to add a HiNative source:
This native speaker says が adds emphasis, and looking at the examples given, it seems like は is a more neutral "I don't have a girlfriend", versus が which says "I don't have a girlfriend, but I have other things".
If anyone has more examples from native speakers, I would be interested in comparing them.]
Yes, you don't use ～を持っています with people, only objects.
[Advanced Topic: Sometimes you can use 持つ with people, such as the noun 彼氏持ち (kareshi mochi), person with a boyfriend, but to say "I have a boyfriend", you should say 彼氏がいます. See HiNative: https://hinative.com/en-US/questions/16793913]