Alignment at equals signs in two columns of an array environment

You can ask TeX to do the computations itself in whatever radix you like (up to 36).

documentclass{article}
usepackage{xparse,array,booktabs}

ExplSyntaxOn

NewDocumentCommand{digitsquares}{sO{c}m}
 {% #1 * option for the header, #2 = alignment, #3 = radix
  IfBooleanTF { #1 }
   {
    tl_set:Nn l_tmpa_tl { multicolumn{6}{c}{mbox{Base ~ $#3$}} \ midrule }
   }
   {
    tl_clear:N l_tmpa_tl
   }
  int_step_inline:nnn { 0 } { int_div_round:nn { #3 } { 2 } - 1 }
   {
    tl_put_right:Nx l_tmpa_tl
     {
      int_to_Base:nn { ##1 } { #3 } ^2 & = & int_to_Base:nn { (##1)*(##1) } { #3 } &
      int_compare:nT { ##1 + int_div_round:nn { #3 } { 2 } < #3 }
       {
        int_to_Base:nn { ##1 + int_div_round:nn { #3 } { 2 } } { #3 }^2 & = &
        int_to_Base:nn
         {
          (##1 + int_div_round:nn { #3 } { 2 })*
          (##1 + int_div_round:nn { #3 } { 2 })
         } { #3 }
        exp_not:N \
       }
     }
   }
  begin{array}[#2]{@{} r @{} >{{}}c<{{}} @{} l @{qquad} r @{} >{{}}c<{{}} @{} l @{}}
  l_tmpa_tl
  end{array}
 }

ExplSyntaxOff

begin{document}

[
digitsquares*[t]{10} qquad digitsquares*[t]{7}
]
[
digitsquares{16}
]

end{document}

The * means that we want the header; the optional argument is for passing the alignment option to array. The mandatory argument is the radix.

The array preamble consists of two sets

r @{} >{{}}c<{{}} @{} l

that is: right aligned column, no space, a centered column with {} on either side of the item (it will be the = sign), left aligned column; in between a qquad separation. The leading and trailing @{} nullify the padding on either side of the array.

I am using a platform that apparently does not allow all the options for an array environment. It may only allow l, r, c, and @{qquad}. I have modified my code so that it gives me exactly what I want.

documentclass{amsart}
usepackage{amsmath}

begin{document}

$$begin{array}{r@{}l @{qquadqquad} r@{}l}
0^2 &{}={} 0, & 5^2 &{}={} 25, \
1^2 &{}={} 1, & 6^2 &{}={} 36, \
2^2 &{}={} 4, & 7^2 &{}={} 49, \
3^2 &{}={} 9, & 8^2 &{}={} 64, \
4^2 &{}={} 16, & 9^2 &{}={} 81.
end{array}$$

end{document}

You have declared a fifth column in between the two groups. In my opinion, it's unnecessary. Here is a simpler code:

documentclass{amsart}
usepackage{amsmath}
usepackage{array}

begin{document}

[
begin{array}{r@{{}²={}}l!{qquad}r@{{}²={}}l}
0&0, & 5&25, \
1&1, & 6&36, \
2&4, & 7&49, \
3&9, & 8&64, \
4&16, &9&81.
end{array}
]

end{document}