This is pretty straightforward. If Ukraine decides to negotiate peace with Russia, the US cannot stop them unless they invade Ukraine or somehow cause a regime change. As in the linked question, this is because Ukraine is a sovereign country that can (in theory) make their own decisions.
However, given that Ukraine is heavily in favor of continuing the war, it's reasonable to suppose that if continuing the war is the US goal, then the US does not need to force Ukraine to continue the war. Ukraine will do it on their own accord, as long as they have the means to do so.
Accordingly, the US can "force Ukraine to not negotiate peace" by providing them with weapons. It's not exactly the same (because the US isn't forcing Ukraine to do anything), but the result is the same - the war goes on.
About the quote in your question: the ultimate reason the war is continuing is that Ukraine and Russia's desired end states are incompatible. If there is peace negotiations (as opposed to an outright victory for one party), chances are both parties will have to compromise to reach a mutually acceptable end state. Ukraine can push for a more desirable end state if they have US support than if they don't. For example, if Ukraine can put "the US will agree not to expand NATO any further if you agree to ____" on the bargaining table, then they can reasonably demand something substantial in ____. If the US does not provide this kind of support, then because Ukraine can demand less, they might feel that continuing the war is their best option.