I am from southern Maine and have a camp in northern Maine. There is a TON of land in northern Maine owned by paper and lumber companies that allow public access - literally hundreds of thousands of acres. I have definitely seen guys running bear hounds up there. I would imagine you could do the same for bobcat but I don't know all the rules and regs for hunting with dogs... If you're looking for a DIY trip, it is definitely doable.
PM me if DIY is what you had in mind and you want more specifics on where.