What the Fallacy of claiming to never be wrong, or being perfect, or always being right?

So prejudice is when you form opinions not based on reason. Putting aside the "I say I'm not therefore I'm not" nonsense, to finish up with a conclusion predicated with "I just think" sounds to me like something which isn't based on reason. 

This is the language of a bigot who can't support their claims but voices them loudly regardless.

Thinking you're always right, perfect, never wrong, or anything along those lines sound more like overconfidence (and a God complex) than any specific fallacy.

Everyone has irrational (cognitive) biases, so I'd be skeptical of someone saying they didn't.

If you claim "X is true because I'm never wrong", that's an unsupported claim ("I'm never wrong" is not evidence).

If someone were to cite this 'never wrong' person as evidence for a point, it could be Appeal to False Authority.

"What the Fallacy of claiming to never be wrong, or being perfect, or always being right?"

Its called Gaslighting, or lying.

"I'm prejudiced only if I hold irrational biases."

That might be a Definist Fallacy. 

"But I don't hold any irrational biases."

This conclusion is derived from his false premise definition which is circularly reinforcing his preconceived assumptions. 

"I just think this country's being overrun by illegal immigrants who are criminals."

Humm, I wonder where he/she got that idea from? The data does not support that claim, statistics show that natural born citizens are more likely to be criminals. Statistics also show that illegal immigrants are more likely to be fleeing from criminals.

If he/she tries to make a follow up claim that crossing the border makes them criminals, and therefore they are all criminals then that could be a circular argument as well.

It might also be an equivocation fallacy as the word is being equated to more hardened criminals, but falls back on crossing a border. It might also be a scapegoat fallacy too, as it attempts to scapegoat the people crossing the border to justify a wall or justify their political party getting into power.