All Purpose Objection

Check out Rights To Ought Fallacy .

"It's my right" and "It's not illegal" are just claims. To qualify as a fallacy, we need more context to see if poor reasoning is involved. For example:

"Dude, you can't take a right on red here!"
"It's not illegal."

This is perfectly fine. However,

"Dude, you shouldn't be mean to that old lady!"
"It's not illegal."

This is fallacious (a Non Sequitur or even a Strawman Fallacy ). The response is not a reasonable response to the "argument."

LessWrong refers to things like these as  'fully generic counterarguments'. You can use them to brush aside all opposition to your position, without having to engage in anything that takes cognitive effort - like considering the very real possibility that you're horribly wrong.

1. ‘It’s my right

There is No Fallacy here unless this is used as a justification in an argument. 

Logical Form:

I want to do / am doing activity X.

I have a right to do X.

Therefore I ought to do X.

This is the Rights To Ought Fallacy. Just because you have a legal right to do something, does not mean it is a good thing to do in the present moment.

2.It’s not illegal.

Again, there is No Fallacy unless you use this to defend a claim or position.

Logical Form:

I want to do / am doing activity X.

X is not illegal / is legal.

Therefore I ought to do X.

This is a blatant Appeal to the Law, or Argument from Law. It is a fallacious authority appeal because laws do not always align with what is rational or moral in a given situation. A dictator who expands/removes term limits is technically governing legally, but risks oppressing their people in the process.

As posed, items 1 and 2 are assertions that are not fallacious without some context.

I think the answer would be different for these two assertions. In the second, asserting that an action is not illegal is usually in response to a different question - if you ask "Is this legal?" then saying it is not illegal is a statement of fact, whether the fact is right or wrong. If you ask "Should we be doing this?" or, "is this right, or ethical, or moral?", then saying it is not illegal is clearly the logical fallacy Moving the Goalposts .

The first statement "It's my right" is more nuanced. This is an assertion of fact being made on the basis of either written law or personal beliefs. If you state "I have a right to do X" when there is no written law or constitutional guarantee of that right, then you are saying it should be your right, or it is a natural right. Your acceptance of the assertion depends on your belief that rights are natural and are being unnaturally proscribed, or, in the alternative view, rights are only granted, by humans, and unless specifically granted they don't really exist as rights. You may think you have the right to "free" speech, but this is not a natural right - it is prescribed in the Bill of Rights, and is therefore subject to limitations and interpretation. At any rate, I see this statement as a disputable statement of fact, not an argument, so it would not be a logical fallacy. IMHO.