The our is lexically scoped. Move it out of the if.

I thought this meant that the variable $bin1 I declared was not global

Quite the opposite, our means you want to use the global variable. There's no reason to use our here. There's almost never a reason to use it. You want to create a variable, so you should be using my.

my $bin1;
if ($binc1 != 0) {
   $bin1 = 1;
} else {
   $bin1 = 0;
}
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The else clause is executed when $binc1 isn't not equal to zero. Avoiding the double negative:

my $bin1;
if ($binc1 == 0) {
   $bin1 = 0;
} else {
   $bin1 = 1;
}
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But you could just simplify instead:

my $bin1;
if ($binc1) {
   $bin1 = 1;
} else {
   $bin1 = 0;
}
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Simplified further:

my $bin1 = $binc1 ? 1 : 0;
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Since the purpose is to format the bit for output, we actually want a string:

my $bin1 = $binc1 ? '1' : '0';
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or

print "The number in binary is: ", ($binc1 ? '1' : '0'), "\n";
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Update: Fixed problem identified in reply.