There is little from a first amendment point of view that is wrong with this bill. The government has broad power to proscribe speech for government employees in their capacity as government employees (if the bill were to mandate the firing of teachers who post pro-gay comments on social media, even if done outside of working hours, that would almost certainly be found unconstitutional).
Where it will run into constitutional problems is its vagueness and blatantly malicious motivation. It is a fundamental principle of law that while "ignorance is no excuse", the government does have a duty to inform its citizens of the law, or, at least, make it theoretically possible for citizens to know the law if they spend enough effort and/or money for lawyers. Much of this bill uses vague terms which a reasonable person would have trouble knowing for sure what they mean. Allowing a teacher to be sued because they had a different interpretation than a judge or jury would be highly problematic from a constitutional point of view.
On the malice front, the bill is written to be facially neutral, but it's clearly not written with a neutral intent. Our culture is replete with references to sex/gender and heterosexual norms. We have different honorifics based on gender (Mr. versus Mrs.), different pronouns, different bathrooms, separate PE classes, etc. The proponents of this bill might argue that these are based on sex, rather than gender, but that would be a difficult argument to make, and should it fail, the bill would leave schools prohibited from making any reference at all to gender; if schools were only to retain cisgender normative references to gender, while prohibiting trans people from expressing their gender identity, that would be clearly discriminatory.
And even if they succeeded in arguing that "gender" in the bill doesn't include sex-based social norms, that leaves references to sexual orientation. The position of "First Lady" is held solely on the basis of being in a heterosexual marriage with the president (that is, every First Lady in the history of the country has held the position solely on that basis; perhaps in the future there may be a president who is not a man married to a woman). I saw a teacher give a lesson during which the students each wrote a poem, and the teacher then suggested to a male student that he might give the poem to a girl. Some teachers talk about their spouse during class. School dances often result in opposite sex partners dancing. And so on.
This bill would prohibit any discussion that might touch on heterosexuality. Any time a teacher makes any comment that suggests that they are heterosexual, or that they anticipate their students being heterosexual, they can be sued. Such a suit must either find in favor of the plaintiff, reject the bill wholesale, or find that the bill only prohibits discussions of homosexuality, with the latter giving very clear grounds for a suit asking for the bill to be declared unconstitutional.
A relevant lawsuit is that challenging Colorado's Amendment 2. This amendment to the Colorado constitution prohibited cities from passing laws against discrimination on the basis of sexual orientation. This amendment was, arguably, facially neutral: although no law prohibiting discrimination against gay people was allowed, laws making discrimination against straight people illegal were equally prohibited. However, this obviously would not impact gay and straight people equally. The Supreme Court found that this amendment restricted the ability of gay people to seek legal protections, and unconstitutionally took away their right to equal access to legislative action. It singled out gay people as being unable to petition their government for the same remedies that were made available to black people, disabled people, women, etc. (Thus, the Supreme Court prohibited the state of Colorado from prohibiting cities from prohibiting businesses from prohibiting employees from engaging in homosexuality. It's confusing to talk about.)
The Florida bill shares many attributes of Amendment 2: it's a facially neutral bill that strips both gay and straight people equally of the platform of public schools, but the larger societal context of the bill makes it discriminatory.