If you do not check yes, there is a good chance they will not consider you for the new job as you obviously have a "problem" with your current job. I have rarely come across such a box; without the box they can call whomever they wish/whenever they wish; it may look like they are being thoughtful by asking if they can call or not but it just raises a red flag if you do not check the box.

Many places do not have a straightforward line of command and/or one can get along with someone over one's immediate boss's head or someone in personnel, etc. so the line on the application form about who your "boss" or who they should call, etc. you give a friendly vice president's name or owner of the company or good friend who trained you, someone other than the person you are arguing with.

I don't care if they start searching for a replacement, they will probably not lay you off or fire you, they will assume you are looking for a job and just be getting themselves in a better place for when you give notice of leaving. You are already trained and they have no basis for replacing you unless you literally are leaving. Yes, they could make it unpleasant so you want to leave even more, but that's why you are looking at other places/leaving in the first place. It just confirms your opinion of them, that they are not where you want to work anymore. On the other hand, people they talk to could be alarmed because they really like you and think you're great and don't realize there's a problem and work to correct the problem so they don't lose you.

If I had a boss I didn't like I'd go to someone higher up on the food chain and tell them I was thinking of leaving and why and what I would like (different department/boss, more money, additional training/consideration to become a supervisor, etc.). Assuming what is going on in the organization and what other people are thinking/know can be a bad idea if you are not interested in leaving anyway for your own/other reasons. If you are interested in leaving; what anyone thinks of you and your decisions, one doesn't really care? No one gets bad-mouthed anymore because there's too much fear of being sued; that's how losers become one's bosses in the first place sometimes, no one investigates too closely or gives too much "useful" information; just don't lie about your salary and how long you worked a place.