Soldiers are federal property. So you could argue they are slaves in a sense and many of the things that are rights to us are privileges to them and could be reduced. But not to the extent you think. See you don’t have control over your property. At the end of the day it’s not yours the right comes from the government.
For example your pet dog is your chattel property. But you can’t just beat or kill it for no reason.
Aside from obviously not being human, dogs are pets, not slaves. Slaves and pets are not necessarily the same category (and thank Christ for that, because the idea of the US military being
pets of the people running the country and not merely their slaves sounds like the fantasy of some deranged anarcho-Communist gooner). Also, in the case we are discussing - that of whether or not soldiers are federal property - they aren't 'my' property or the property of any single individual, but the (federal) government directly. So I don't think individual property rights and the question of where they come from are particularly relevant in that scenario, especially if your answer to the latter is 'the government'.
Anyway, ordinarily I would define modern US military personnel as voluntary citizen-soldiers and employees of the state, rather than property and therefore slaves (unlike, arguably, conscripts for example). But sure, let's roll with this definition of the US serviceman as chattel owned by the federal government, at least for the duration of his contract (which would make him an indentured servant, IOW, a sub-category of slave as I have said). I can buy that in light of Biden's government forcing the COVID vaccine on the entire US military and giving the boot to anyone who won't take it.
I would argue that forced vaccination is a far more severe breach of the individual soldier's rights & personal autonomy than looking up their search history could ever be, considering that not only does it cross into the physical realm but the vaxx is by now well documented as causing all sorts of adverse effects (up to death or being crippled for life) even in otherwise young & fit individuals for dubious benefit, as even fully-vaxxed soldiers have been reported to die of the coof anyway. Yes it might be epically embarrassing if you get dishonorably discharged because your superior officer found out you were posting online about how much you hate America, want it to be destroyed, and spit on the graves of other soldiers who died recently in-between jacking off to horse/loli gangbang hentai or whatever nightmare gets the pornsick brains of Burning Bushnell and his ilk's going, but at least you won't die or be paralyzed for the rest of your life. Can't say the same about the vaxx.
Suffice to say that any argument that the former is OK while the latter is somehow too egregious a violation of the average soldier's rights to be countenanced, is one that I can't take seriously and which at my most charitable I'd view as one of hilariously misguided & skewed priorities - the equivalent to treating murder as a misdemeanor and catcalling as a high crime worthy of the death penalty. Hell you could just as easily make an argument to justify the latter on the same health grounds as the former (if not even more-so), just one of mental health rather than physical. After all, a guy who posts as obsessively about how much he hates himself, his supposed comrades-in-arms, the Constitution and the very ground he walks on as Bushnell did, most probably has a few screws loose. And perhaps he'd still be alive and no threat to said comrades (or anyone else) if he were kicked out of the force and directed to the treatment he clearly was in dire need of, etc., etc.