A EULA cannot force you to give up your constitutional rights.
That...varies. The right to a speedy trial can be waived. The right against enslavement cannot. You can consent to a warrantless or otherwise unreasonable search given adequate specificity, but outside of a condition for release from otherwise-constitutional more severe deprivation of liberty (e.g., parole from prison) I don't think it can be generally waived in advance, only searches specifically and immediately consented to when being executed. But I don't have any particular cases in mind, and that understanding could be wrong. But “constitutional rights can be waived” is definitely way to broad to be a good useful guide for resolving specific isssues.
There are limitations, of course. If you live off-post in private housing, your commander has no legal authority to inspect it. They can only go through your stuff if you live in government-owned housing.
That's not a waiver, that's a space warrantless searches are reasonable under the fourth amendment and Congress’ Art. I, Sec. 8 powers with regard to the military, etc. Otherwise, when we had conscription, conscripts, who do not freely consent, would have been immune.
"In the exercise of his rights and freedoms, everyone shall be subject only to such limitations as are determined by law solely for the purpose of securing due recognition and respect for the rights and freedoms of others and of meeting the just requirements of morality, public order and the general welfare in a democratic society."