The difference is that striking soldiers can cause irreparable harm while the damage done by teachers can be easily repaired.

Let's say some soldiers start striking during a war for better pay. That puts the country in an immediate danger as the enemy advances and potentially causes irreparable harm (a change that cannot be undone). Even if one can retake the positions given up during the strike, many will die doing so. Clearly, there's a problem there.

Now let's say some teachers start striking in September. Accepting Adams' premise that an uneducated populace is a threat to national security, what affect do striking teachers have on national security? Well, maybe those students have class from October through July that year or they loose a few weeks of schooling. The difference is that the harm is not irreparable. The harm is merely a delay or, at absolute worse, a very marginal harm. In fact, any harm is probably about the same as moving from one school district to another. Different schools teach in slightly different orders and in a slightly different way and so when moving schools, a student might loose out on a small portion of learning just as they would during a strike.

Others can argue whether unions are good or bad, but I think there's a clear difference between the irreparable harm that can be caused by a military strike when compared with the reparable harm caused by a teacher strike. So, rather than September to June, students have an October to July school year one year. There's a distinct difference there.

Now, before one says, "well, what about strikes during peacetime", the military operates under a readiness principle. While I don't believe that countries aren't lining up to attack the US, the whole point of a military is to be ready for that possibility. So, it's peacetime, the army is on strike and then someone attacks. By the time you order the strike to stop (due to it no longer being peacetime), irreparable harm could already be done. Striking soldiers wouldn't be on bases doing drill exercises. They might be at home thousands of miles away from. In fact, Israel fought a war somewhat like this. During Yom Kippur, religious Jews don't eat, don't use electricity, and don't work. That's the perfect time to attack - most of the military was home and wouldn't get fast word that an attack had occurred and some wouldn't do anything about it even if they heard for religious reasons. Suffice it to say, a union on strike could provide an attacker with a similar advantage. Soldiers aren't on base, some might not come back if ordered to end the strike, some might not hear about it immediately, etc.

The difference is irreparable vs reparable harm. A strike by a teachers' union can be repaired by teaching a bit into the summer and the harm caused by that lack of teaching already happens to students when they move schools and have a small mismatch in the curricula. I really love the premise that education is essential to national security because I agree that trade and education create stability that means war is very unlikely. I just don't see the harm caused by a teacher strike to be the same as the harm caused by a military strike.