I see two problems here:
The first one is that the difficulty of proving "Bigfoot is not an extant creature living in North America" has nothing to do with the logical form of the statement. Consider the statement "Elephants are not an extant creature living in my room", which has the same logical form, but can be proved without effort.
The difficulties of proving statements of the form ¬(]x)(Fx) stem from the fact that ¬(]x)(Fx) is equivalent to (Vx)¬(Fx) which is an universal statement. Universal statements, if true, are true of everything in the universe, and, ignoring logical truths, this implies that proving them true requires one to make sure that every single object in the universe satisfies the conditions.
Adding a second condition to the universal statement can make this task manageable. "Elephants are not an extant creature living in my room" is easily provable because it's trivially true of almost every object in the universe that they are not in my room, and I can easily check whether any object in my room is an elephant; however, if there's a lot of objects that satisfy the second condition, proving that they lack the first one can be unmanageable.
The second problem I see is that the notion of 'burden of proof' is not really related with the positivity or negativity of statements. For instance, it's generally accepted that that lack of vitamin C causes scurvy. You can deny this claim, but the burden of proof is on you. Also, answering 'who has the burden of proof' seems to be context dependent. It sometimes depend on shared assumptions about whats true, sometimes on power relations, sometimes on the consequences of accepting that a statement is true. Saying 'Bigfoot exists' at a coffee table is, after all, not the same as asking for a research grant to search for Bigfoot.