Is $\Bbb R^2\times\Bbb S^2$ simply connected?

If a space $X$ is simply connected, then each space $Y$ which is homeomorphic to $X$ is also simply connected. A stronger result is that if $Y$ is homotopy equivalent to $X$, then $Y$ is simply connected. See Proving that the fundamental groups of two spaces with same homotopy type are isomorphic .

Now $\mathbb R^2 \times S^2$ is homotopy equivalent to $S^2$. But it is well-known that $S^2$ is simply connected.

You can also invoke the fact that $\pi_1(A \times B) \approx \pi_1(A) \times \pi_1(B)$. But $\pi_1(\mathbb R^2) = 0$ and $\pi_1(\mathbb S^2) = 0$.

I'll try to avoid using $\pi_1$ as it wasn't mentioned in the question originally, although as Paul has mentioned there is a quick proof using $\pi_1$.

There are a few questions embedded in this post, so I'll try to address them one at a time. First and foremost, $\mathbb{R}^4$ with a line excised is simply connected. We can actually prove this using an easy to visualize case. Let's reduce our dimensions by $1$ across the board - let's consider the case of $\mathbb{R}^3\setminus \{p\}$. That is, we take three-space and remove a point. I trust your visual intuition informs you that any loop closes to a point continuously.

Indeed, one can simply imagine moving the loop far away from the point into a region of space that is far from the puncture, and then closing the loop. You can make this rigorous by some arguments using parametrizations (a good exercise).

Now, in the case of $\mathbb{R}^4\setminus \ell$, where $\ell\subseteq \mathbb{R}^4$ is a line it is no loss of generality to assume that $\ell$ is a linear subspace, i.e. passes through the origin. It is also no loss of generality to assume that $\ell=\{(0,0,0,s\}: s\in \mathbb{R}\}$, i.e. one of the axes. We have a loop $\gamma:S^1\to \mathbb{R}^4\setminus \ell$. Now, we can consider the homotopy taking $\mathbb{R}^4\setminus \ell$ to the space $\mathbb{R}^3\setminus \{0\}$ given by $$ F:\mathbb{R}^4\setminus \ell \times [0,1]\to \mathbb{R}^4\setminus \ell$$ and $F((w,x,y,z),t)=(w,x,y,tz).$ At $t=1$ this is the identity map, while at $t=0$ this is the projection onto $\mathbb{R}^3\setminus \{0\}\subseteq\mathbb{R}^4\setminus\ell.$ In particular, this homotopy slides the loop $\gamma$ into a loop in $\mathbb{R}^3\setminus \{0\}$. The loop in $\mathbb{R}^3\setminus \{0\}$ is then clearly contractible by what we have said above.

Of course, this was not necessary. Once we know that $\mathbb{R}^2\times \mathbb{S}^2\cong \mathbb{R}^4\setminus \ell$, we can conclude that both are simply connected or neither because simple connectedness is preserved under homeomorphism. To see that $\mathbb{R}^2\times \mathbb{S}^2$ is simply connected, just use an idea similar to the above to exhibit a homotopy $\mathbb{R}^2\times \mathbb{S}^2\simeq \mathbb{S}^2$.

From the point of view of Category theory, $π_1$ is functorial, and has a right adjoint, the classifying space functor. As a result, it respects products. Thus $π_1(\Bbb R^2×S^2)\congπ_1(\Bbb R^2)×π_1(S^2)\cong e×e\cong e$.