Why is the L2 penalty squared but the L1 penalty isn't in elastic-net regression?

There was some data set I worked with which I wanted to solve non negative least squares (NNLS) on and I wanted a sparse model. After a bit of experiementing I found that what worked the best for me was using the following loss function:

$$\min_{x \geq 0} ||Ax-b|| + \lambda_1||x||_2^2+\lambda_2||x||_1^2$$

Where the L2 squared penalty was implemented by adding white noise with a standard deveation of $\sqrt{\lambda_1}$ to $A$ (which can be showed to be equivelent to ridge regression in the expectation) and I implemented the L1 squared penalty by adding a row to $A$ with a constant value of $\sqrt{\lambda_2}$ and added a value of 0 to the end of $b$, which in the case of NNLS can be shown to be equivelent to L1 squared penalty.

That worked good for my purposes, but I know that usually in sparse regression models (for example elastic net or lasso regression) the L1 penalty is not squared, so it made me wonder if there could be a problem I'm missing with using squared L1 penalty? Is there any specific reason the L1 penalty is not squared but the L2 penalty is squared in elastic net regression?