Above is some information you may have seen, but it's not necessarily legally correct. Below is some information you may want to consider:

As far as invasive testing procedures go:

The California Supreme Court has allowed refusal of medical treatment, as a constitutional right of privacy, per Cal Const Art I § 1; and the U.S. Constitution 5th and 9th Amendmendments’ guarantees to the individual the freedom to choose to reject, or refuse to consent to, intrusions of his or her bodily integrity, including medical treatment. Bartling v. Superior Court (Cal. App. 2d Dist. Dec. 27, 1984), 163 Cal. App. 3d 186, 209 Cal. Rptr. 220.

There can be an exception if the employee has already signed some sort of “consent” to medical treatment form as a condition of employment. In one case where the employee then refused the medical treatment once they were hired, it was held that the consent was valid, thus refusing to abide by an employment agreement was a rightful reason for termination. Feminist Women's Health Center v. Superior Court (Cal. App. 3d Dist. Jan. 31, 1997), 52 Cal. App. 4th 1234, 61 Cal. Rptr. 2d 187, 1997 Cal. App. LEXIS 117.

Thus, if your employment agreement contained some sort of clause by which you already consented to an additional testing for viruses like COVID, it may be difficult(but not impossible) to fight it. 

The notice you see at the far top says a federal judge ruled on the topic. However a federal judge cannot overrule a state supreme court decision. 

In the United States, a state supreme court (known by other names in some states) is the highest court in the state judiciary of a U.S. state. On matters of state law, the judgment of a state supreme court is considered final and binding in both state and federal courts. Decisions on state supreme court cases can be appealed to the Supreme Court of the United States if they involve a point of federal law.

Also consider this:  The state is planning on forcing employees to sign a consent form right before testing. If they are threatening you with discipline or termination unless you sign the consent waiver, it's not really consensual.

However, even in regular situations where the employer is allowed to require some sort of medical screening for their potential employees(like drug tests, regular medical treatments before or during employment), if it is not done in the LEAST invasive way possible, it can still be found unconstitutional. (Under Cal Const Art I § 1, directly observing a correctional officer’s urination during a drug test violated her right to privacy. Hansen v. California Dep't of Corrections (N.D. Cal. Mar. 25, 1996), 920 F. Supp. 1480, 1996 U.S. Dist. LEXIS 11361.)  (To pass constitutional muster, the intrusiveness of the testing must be weighed, along with an individual’s legitimate expectation of privacy, the nature and immediacy of the government concern at issue, and the efficacy of drug testing in meeting that concern. Deborah M. v. Superior Court (Cal. App. 4th Dist. Apr. 29, 2005), 128 Cal. App. 4th 1181, 27 Cal. Rptr. 3d 757, 2005 Cal. App. LEXIS 681.) 

If you are opposed to being forced to take the mandatory covid-19 testing you can contact Liberty Counsel at Liberty@LC.org

They are a very high profile litigation firm that do not charge as they are a non profit organization. 

The state supreme court has already ruled on invasive medical procedures and tests. A federal judge can't alter that decision.