The use of squared slack variables is well-known in nonlinear programming, making possible to convert a problem with inequality constraints into a problem containing only equality constraints. It is an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase of the dimension of the problem, the numerical instabilities, and the singularities. The situation changes, however, in nonlinear second-order cone programming, where the strategy has a reasonable advantage. The reformulated problem with squared slack variables has no longer conic constraints, which enables us to deal with the problem as an ordinary nonlinear programming problem. In this work, we establish the relation between the Karush-Kuhn-Tucker points of the original and the reformulated problems by means of the second-order sufficient conditions and regularity conditions.