Translation:My name is John.
That's an interesting idea, but there are a few problems with your suggested usage.
First and foremost, the fact is that you would never really "correct" someone's choice of honorific. You don't get to choose their view of their relationship to you, which is largely what honorific choice represents. Sure, you, and all Japanese people, will have expectations of which honorific people should use, based on your view of the relationship and the unspoken social rules that Japanese people learn (and influence) by growing up in Japan, but ultimately, they are just expectations and typically remain unspoken.
Secondly, while there is no "correct" or "incorrect" honorific choice, the use, or lack, of honorifics can speak volumes about one's social status. For instance, the use of さん indicates that the speaker is showing some level of respect to that person, putting them "higher" than the speaker. Note that this is the baseline level of respect strangers of roughly equal social standing extend to one another in civil conversation, since the default stance is to be deferential. So, insisting to be called ～さん is tantamount to insisting to be shown respect.
All of what I've said so far is talking about why Japanese people generally don't insist on a particular honorific, but it can be, and sometimes is done. However, the most incongruous part of your suggestion stems from the nature of もうします. This verb is from a part of speech called kenjougo, or "humble language", which is used to show respect to the listener by putting the speaker below them. So, it doesn't make sense to insist someone calls you ～さん, i.e. places you above them, in a way that places yourself below them.
There are myriad alternatives for how to insist on a particular honorific, depending on exactly how strongly you want to insist, your actual relative social standing, relative age, gender, how passive-aggressive/rude you want to sound, and so on. Whether or not と gets used depends on the verb you choose, and/or how you choose to structure the sentence.
No, unfortunately that's not how と works. As others have pointed out, it's more of a quoting particle, so you're saying "I am called "John". "
Other ways it is used include describing one's thoughts (～と思います【おもいます】= "I think ～") and quoting others' speech (ジョンさんは～と叫びました【さびました】= "John yelled ～").
It's also used in more advanced ways with specific verb forms, e.g. 話そう【はなそう】とします = "to try to speak", but those cases are very different from this exercise's sentence.
と is sometimes called the quotative particle. It follows the word or phrase that is being stated / declared / quoted. Particles in Japanese are also called postpositions, because they follow the things they mark or identify, in contrast with our prepositions, that are (preposed) put in front of their objects.