People have provided several workable alternatives to the original sentence, which in its original form seems ambiguous. Does it mean that each individual (each boy and Jen) has three balloons, or that together, Jen and an unidentified number of boys have three balloons? That ambiguity can be cleared up in a number of ways, including several of those suggested, or by saying "The boys and Jen have three balloons each."
The answer to the question of which verb form to use is clearer. It's actually the conjunction ("and") that determines this.
Here's the rule I learned. When the conjunction is "or", the noun that follows "or" and is closest to the verb determines which verb to use. The result, though correct, sometimes sounds strange. Had the original sentence begun with "Every boy or Jen", the correct verb would have been "has", because "has" is the proper verb form to use with the singular "Jen" (or with "every boy" for that matter).
But when the conjunction is "and", as it is in the original sentence, the rule is to assume a plural subject and to use the appropriate verb form to go with it. So the use of "have" in the original sentence is correct.
So regarding your question about the original sentence, I'd say that because of the presence of "and", there's no doubt about the verb "have". But the wording of "Every boy and Jen", while not strictly incorrect, could be much clearer.