No, you don’t, but try to recognize and see how meditation is itself a spiritual activity. You’re diving into this ‘underworld’ realm of the mind, encouraging yourself to experience things that you don’t have any understanding of, and then trying to get some benefit out of it.
To contrast, let’s take something like prayer, which I hope we can agree is de facto a spiritual activity, one you would normally think you’d need to be spiritual in order to do. Let’s break prayer down into a bunch of components. Let’s say you did all of those component activities, without believing that prayer actually does anything.
You got on your knees, put your hands together, looked up towards the God you don’t think exists, and said words that most believers would consider a prayer. Did you just perform a spiritual act? Let’s say something happened in your mind while you did that, you had an experience that millions of people would have labeled a spiritual experience while you did that. Was that experience somehow not a spiritual experience because you didn’t believe it was?
At some point, the mundane meets the spiritual, regardless of whether or not you actually believe in it or not. Doing things like meditating naturally brings you closer to the line. People who don’t believe in any kind of spiritual reality can have crazily-spiritual experiences, and then just call it daydreaming later.
People who are spiritual realize that rejection of belief doesn’t really solve anything about anything. When they have spiritual experiences, they just call them spiritual and don’t try to whitewash them with rationality.
So in short, no, you don’t have to be spiritual, to meditate, but that won’t stop you from having spiritual experiences while you’re meditating. If your mind is set up to have them, then getting close to that line will just cause your mind to have them. And your conscious, thinking mind will have to make sense of them. It’s still up to you how you want to deal with them. Take meaning out of them or don’t, your choice.