Another way to state the implicit argument is "If corporations are considered persons with regards to free speech, then we must be consistent and afford them other rights, such as the ability to run for office." No name-calling necessary.
Although I consider that beside the point, since this company did more than just make an ad. They actually filed to take part in the Republican primary. Assuming they get denied, then that could set precedent for the limits of corporate person-hood.
Unfortunatly in this specific case, no, because as another person pointed out, the corporation in question is not yet 25 years old. The real question will be deferred until a corporation which meets all other requirements runs.
Corporations are not citizens, they do not meet the requirements to stand for election. Similarly, corporations can't cast votes or sit on juries or be drafted into the armed forces.
Although I consider that beside the point, since this company did more than just make an ad. They actually filed to take part in the Republican primary. Assuming they get denied, then that could set precedent for the limits of corporate person-hood.