map({key:.,value:1})|from_entries|keys_unsorted
as an alternative to unique seems to work.
map turns the array into:
[{"key":"cross","value":1},{"key":"base","value":1},{"key":"cross","value":1},{"key":"dunk","value":1}]
from_entries turns that into:
{"cross":1,"base":1,"dunk":1}
Removing the duplicates at that point since two elements of an object can't have the same key.
And keys_unsorted returns the keys of that object, in the original order.
A shorter/simpler alternative working on the same principle:
map({(.):1})|add|keys_unsorted
Or you can use a more generic programming language such as perl:
perl -l -0777 -MJSON::PP -MList::Util=uniq -pe '
$_ = encode_json[uniq @{decode_json $_}]' array.json
-p is the sed mode, pull each record of array.json into $_ for the eexpression to work on.
-0777 sets the record separator to something impossible, meaning there's only one record being the full file (see also -gobble in newer versions of perl).
decode_json $_ decodes that record and returns an array reference, @{...} makes that a list passed to uniq, [...] makes the result another array reference passed to encode_json, the resulting $_ is printed (followed by a newline with -l).
There are a number of JSON plugins for perl. JSON::PP (PP for pure perl) is part of the core of perl, so should always be available. JSON::XS would be more efficient and also comes with a json_xs which is handy to use in shell scripts:
json_xs -e 'use List::Util "uniq"; $_ = [uniq @$_]' < array.json
If you're already familiar with perl, that means you don't need to learn a new language like that of jqs and can reuse what you know to process other formats like XML, YAML, etc.