Based on what's been said so far, the simple answer is no, there is no claim for wrongful termination. The lifeguards were fired, according to the city, for violating terms of their employment agreement that prohibited the use of city property for personal gain. The stated repercussion of doing so was disciplinary action. In most circumstances, disciplinary action can mean anything from a meeting with your boss, written documentation for your employment file and termination, among other things.
Certainly it is possible that there are many more factors at play here and it is therefore impossible to say with complete certainty that the fired lifeguards have no legal recourse against the city of El Monte.
Fourteen lifeguards and a pool manager employed at the El Monte aquatic center lost their jobs over a parody of the pop culture hit "Gangnam Style." While on break, or after work, the lifeguards created a video called "Lifeguard Style" while dressed in their city uniforms and on city property and then posted the video to YouTube.
The involved guards insist that no member of the public was involved or at the pool during the shooting of the video, but the El Monte City Council has so far stood behind the choice to fire the lifeguards. The reputation of the city was compromised, according to city officials, and the rules were broken.
Several members of the public spoke out at a recent City Council meeting requesting that the decision to fire the lifeguards be reconsidered. One important fact that may be a strong reason for rehiring the lifeguards is the lack of experience of their replacements. Former cashiers received three days of training and are now filling the openings left by the discharged lifeguards. Parents complained about the change and concern for their children's safety at the meeting.
Source: Los Angeles Times, "'Gangnam Style' lifeguards: Mayor calls for review of firings," September 19, 2012