I think there are several points to your question. First of all, the link between the declaration of neutrality and the possibility of a civil war. As far as I know, it's one reason but far from being the only one. Yes, French politicians were especially concerned about the possibility of a far-right, anti-parlementarist coup. To understand that position, you need to go back 2 years earlier, Februrary 6th, 1934. That day, a right-wing demonstration turned into an all-out riot that ended up wit 37 deads and more than 2000 injured. The trauma of the civil war-like accounts of that day are among the reasons that helped all the left-wing forces to unite into the Popular Front (Socialists, Radicals, Communists). But that's not the only force that forced Blum's hand. I'd add two other very important factors.

First of all, the pacifist sentiment was strong in France, among civilians and politicians alike. The trauma of the Great War was very real and the majority wanted to avoid another war, at all costs. That's one of the reasons Hitler had such an easy time before 1939, France tried everything to avoid another war with Germany, even if that meant "accomodate" Hitler.

Second, England pressure for neutrality was quite strong. The conservative government of Baldwin used Germany's threat as leverage, and basically threatened France to withdraw its support if the PF were to intervene in favor of the Spanish Republicans. At this point, France was in quite the economic turmoil and couldn't pass on british economic and political support.

As Jackson or Wendell have shown, the concern of a civil war or a Coup was very real at that time. Once again, the trauma of 1934 was still alive, and many of the right wing and far-right wing groups were praising Franco and citing him as a model. Since some of those groups were very connected to veterans and the army in general (Most notably les Croix de feu, but there were other very influential groups such as the royalist and antisemitic Action Française or the Jeunesse Patriotes), people could rightfully fear a right-wing militaristic coup. In addition, the 3rd Republic had already be confronted to such a perspective during the Boulanger movement during the late XIXth century.

Now from a modern perspective, it's hard to say if the possibility was real or not. I wouldn't venture into prospective opinions about the realism of a coup at that point of time. But you should note that the society and the political landscape was very polarized, especially after the Dreyfus case, where both sides of the issues seemed irreconcilable during all the decades that led to WW2. Political violence was also a landmark of the 3rd republic, whether we're talking about far-rights ligues or anarchists bombings. Lastly, the international context could be seen as favorable to such a regime change, since France was, at that point, essentially surrounded by militaristic authoritarian (or even totalitarian) regimes that inspired a lot of French actors.

Sources : Jackson, G. The Spanish Republic and the Civil War, 1931 - 1939, 1987

Windell, George G. "Leon Blum and the Crisis over Spain, 1936" The Historian; Kingston, R.I., etc. Vol. 24, N° 4, (Aug 1, 1962): 423.