Their statutory mission doesn't preclude them from questioning, detaining, or arresting US citizens like any other law enforcement. 8 USC 1357 specifically grants them arrest power that are not specific to immigrants.
§1357. Powers of immigration officers and employees
(a) Powers without warrant
Any officer or employee of the Service authorized under regulations prescribed by the Attorney General shall have power without warrant—
(1) to interrogate any alien or person believed to be an alien as to his right to be or to remain in the United States;
(2) to arrest any alien who in his presence or view is entering or attempting to enter the United States in violation of any law or regulation made in pursuance of law regulating the admission, exclusion, expulsion, or removal of aliens, or to arrest any alien in the United States, if he has reason to believe that the alien so arrested is in the United States in violation of any such law or regulation and is likely to escape before a warrant can be obtained for his arrest, but the alien arrested shall be taken without unnecessary delay for examination before an officer of the Service having authority to examine aliens as to their right to enter or remain in the United States;
(3) within a reasonable distance from any external boundary of the United States, to board and search for aliens any vessel within the territorial waters of the United States and any railway car, aircraft, conveyance, or vehicle, and within a distance of twenty-five miles from any such external boundary to have access to private lands, but not dwellings, for the purpose of patrolling the border to prevent the illegal entry of aliens into the United States;
(4) to make arrests for felonies which have been committed and which are cognizable under any law of the United States regulating the admission, exclusion, expulsion, or removal of aliens, if he has reason to believe that the person so arrested is guilty of such felony and if there is likelihood of the person escaping before a warrant can be obtained for his arrest, but the person arrested shall be taken without unnecessary delay before the nearest available officer empowered to commit persons charged with offenses against the laws of the United States; and
(5) to make arrests—
(A) for any offense against the United States, if the offense is committed in the officer's or employee's presence, or
(B) for any felony cognizable under the laws of the United States, if the officer or employee has reasonable grounds to believe that the person to be arrested has committed or is committing such a felony,
if the officer or employee is performing duties relating to the enforcement of the immigration laws at the time of the arrest and if there is a likelihood of the person escaping before a warrant can be obtained for his arrest.
Under regulations prescribed by the Attorney General, an officer or employee of the Service may carry a firearm and may execute and serve any order, warrant, subpoena, summons, or other process issued under the authority of the United States. The authority to make arrests under paragraph (5)(B) shall only be effective on and after the date on which the Attorney General publishes final regulations which (i) prescribe the categories of officers and employees of the Service who may use force (including deadly force) and the circumstances under which such force may be used, (ii) establish standards with respect to enforcement activities of the Service, (iii) require that any officer or employee of the Service is not authorized to make arrests under paragraph (5)(B) unless the officer or employee has received certification as having completed a training program which covers such arrests and standards described in clause (ii), and (iv) establish an expedited, internal review process for violations of such standards, which process is consistent with standard agency procedure regarding confidentiality of matters related to internal investigations.
Relevant sections highlighted. (a)(1) specifically allows questioning of any person they suspect is an Alien. Courts generally give officer judgement in this a wide latitude. Section 5(a) and (b) is where they get the power to enforce laws against anyone. Provided it happens in their presence and while they are otherwise on official duty. Further, the section immediately following gives them the power to "execute and serve any order, warrant, subpoena, summons, or other process issued under the authority of the United States. "
They are Federal law enforcement and believing social media saying you can ignore them is bad advice. Exercise your rights the same as you would with the FBI, but don't think just because they are are ICE and you are a US citizen means they can't question you, detain you, or arrest you. They can.
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