Let $\gamma:[0,1]\to \Bbb R^2$ is a smooth curve and $(a,b),(c,d)\in \Bbb R^2$ are two points. To obtain the shortest distance of $(a,b)$ and $(c,d)$ from the curve (simultaneously) we must minimize the following: $$Dis=\sqrt{(a-\gamma_1(t))^2+(b-\gamma_2(t))^2}+\sqrt{(c-\gamma_1(t))^2+(d-\gamma_2(t))^2},$$
Can one minimize $(a-\gamma_1(t))^2+(b-\gamma_2(t))^2+(c-\gamma_1(t))^2+(d-\gamma_2(t))^2$ instead of $Dis$? my second question is to obtain the Minimum of $Dis$ we must solve two ODE with at least two free parameter (constant). How to obtain this two constant?
To answer your first question: no. Consider some curve where the shortest distance would be $5$, with distance $2$ from the first point and $3$ from the second. Also suppose that at another moment the distance would be $2.5$ from the first point and $2.55$ from the second. This is not the shortest distance of course, but as for the sum of squares we would get $4+9=13$ and $2.5^2+2.55^5=12.7525<13$, so here it is not the shortest distance.