In particle in square potential barrier problem, we can easily find that some probabilities exist which express how many particles can go beyond of the potential wall.
As there may be a language problem here, there are two interpretations of your question.
1) the barrier problem which allows for tunneling
2) The square well potential which will give energy levels for occupation
So my question is that, can we find some particles in the potential wall - square of wave function's norm really means probability of find a particle in the wall?
Answering 1), the answer is yes.
It is called tunneling, and there is a quantum mechanical probability for the particle to exist beyond the barrier.
Answering 2) the link given by Qmecanic has a number of relevant answers . which also say yes for a potential well if the energy of the particle is an energy level of the well and is unoccupied.