Neither a felony assault charge nor conviction prevents someone from running or being elected El Presidente. In fact, it might allow a candidate to play up military credentials.
Straight from ye olde wikipedia:
Article II, Section 1, Clause 5 of the Constitution sets the following qualifications for holding the presidency:
be a natural-born citizen of the United States;[note 1]
be at least thirty-five years old;
be a resident in the United States for at least fourteen years.
The Twelfth Amendment precludes anyone ineligible to being the president from becoming the vice president.
A person who meets the above qualifications is still disqualified from holding the office of president under any of the following conditions:
Under the Twenty-second Amendment, no person can be elected president more than twice. The amendment also specifies that if any eligible person serves as president or acting president for more than two years of a term for which some other eligible person was elected president, the former can only be elected president once. Scholars disagree over whether a person precluded by the Twenty-second Amendment to being elected president is also precluded to being vice president.[69]
Under Article I, Section 3, Clause 7, upon conviction in impeachment cases, the Senate has the option of disqualifying convicted individuals from holding federal office, including that of president.[70]
Under Section 3 of the Fourteenth Amendment, no person who swore an oath to support the Constitution, and later rebelled against the United States, can become president. However, this disqualification can be lifted by a two-thirds vote of each house of Congress.
Now, if the candidate was in an office that allows them to be impeached by the Senate, it's possible that that august body could lay down the "no running for Federal Office" smackdown. I don't believe that any of the still running candidates are in that line of fire, though.