The minimum such N is 1225. Its own square root is 35, while sqrt(7225) is 85.

guide=1;

found=false;

while ~found

  guide=guide+1;

  ones=ceil(sqrt(10^guide)):floor(sqrt(2*10^guide));

  sevens=ceil(sqrt(7*10^guide)):floor(sqrt(8*10^guide));

  ones=mod(ones.^2,10^guide);

  sevens=mod(sevens.^2,10^guide);

  int=intersect(ones,sevens);

  if length(int)>0

    disp(int)

    for i=1:length(int)

    disp([sqrt(10^guide+int(i)) sqrt(7*10^guide+int(i) )])

    end

    disp(' ')

    if guide>5

      break

    end

  end

end

finding

                         225
                    35    85
     
                        5625
                   125   275
     
           22500       48225
       350   850
                   385   865
 
          155625      562500
1075        2675
            1250        2750

        

The first line of each group shows the portion of the number without the leading digit (the 1 or the 7). The larger cases have two listed. The remaining line(s) each have the square root of the 1 version and the square root of the 7 version; in the case of two separate "remaining digits" on the same line, separate lines are used for each pair of square roots.