I'm inclined to say, "you don't have to agree that 'gender is a social construct'." Of course, you don't have to agree that the Earth revolves around the sun, either. But people who spend large chunks of their lives to studying either tend to agree with those statements.
The thing is, most of the things we talk about are social constructs. E.g., languages are social constructs. My own, highly non-technical definition of a social construct is that it's something that only exists because humans agree/believe/act like it exists. English exists only to the extent that people speak it, read it, write it.
A social construct can be one's identity. Millions of USAans identify as American. They consider it an essential part of who they are and they feel very uncomfortable when they are asked to hide or even just tone down any part of their American-ness. I don't particularly identify with being "American," yet when I lived in Europe for several years, I kept running into the myriad ways that Americanness had become a part of my self (the way Strontium-90 becomes a part of one's bones.) Yet "American" didn't exist 500 years ago, and students of American history and literature have traced how people in what is now the USA created and constructed this concept.
The thing is, people start out without labels. They're just the way they are. (If you've had a couple of children, you've noticed how different they can be right from the minute they pop out.) But then society comes along and says "this is a boy and boys are like this, and that is a girl and girls are like that." And that everyone is either one or the other (not both, not neither, not sort-of anything. You're all M or you're all F.) And this indoctrination starts at birth. They learn What Girls Are the way they learn What Cats Are. Plus, they learn what society says they are. But if there's a conflict between the way a child actually is (something the child can experience directly) and what they say their assigned gender is, then they can either (a) try to turn themselves into something they aren't or (b) take on the entire adult world and tell them they're wrong.
What's interesting to me is the ever-growing number of people who identify as non-binary. To me, this is, if anything, evidence for the arbitrariness and thus the constructedness of gender. If "being a boy" and "being a girl" were something built into people, rather than a classification based on socially defined concepts, we wouldn't expect to find people who are genderfluid, agender, mix-and-match, etc. I think there have always been people like that, they just had to describe themselves in the terms and fit into the roles society supplied them with, while knowing that those terms didn't really fit.
