Is it true? Ask Jelly!

Jelly, 3 bytes

ṭFẠ

F flattens the input list.

ṭ tacks on the original input list as an element, which is falsy if and only if it is empty.

Ạ then checks if any element in the flattened list, or the original list itself, is falsy.

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(Original answer)

FẠ^Ṇ

Thanks to Dennis for encouraging finding a solution matching his.

FẠ gives 0 if the input contains a falsy value at any depth, else 1. This is what Ȧ does, except for empty lists.

Ṇ gives 1 if the input is a falsy value, else 0. The only falsy list is the empty list.

XOR-ing the two gives the answer.

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F;WẠ

This is much in the same spirit as Dennis's F;LẠ, but instead of using L to put a zero in the list when the list is empty, it uses W to put the empty list into itself (producing [[]]), making it contain a falsy element.

Retina, 10 bytes

A`\b0
^...

Try it online!

First we remove the input if it contains a zero. The we try to match at least three characters from the beginning of the string (to ensure that the input hasn't been eliminated in the previous stage, or was only [] to begin with).

Ruby, 25 24 23 18 16 bytes

p$_!~/\D0|^..$/

Requires the -n flag on the command line (+1 byte, -e -> -ne).

Try it online!

This is a full program that takes input in Ruby's canonical array format on STDIN and outputs true or false on STDOUT.

 $_              # line of input that was read automatically (-n)
   !~/        /  # does not match the regex...
      \D0        #   a non-digit followed by a 0
         |       #   or...
          ^..$   #   a 2-length string (which must be [], the empty array)
p                # output the result

23 byte function version:

->a{"#{a}"!~/\D0|^..$/}

This is a proc that takes one argument, the array to be tested.

Thanks to Martin Ender for a byte and to Ventero for two bytes!