Sure! Your provided input is not valid JSON, but I’m going to assume the trailing commas on those objects are removed and the objects are wrapped in an array, which would be the root object of the JSON document.
First, we can transform the millisecond-precision UNIX dates into second-precision, which is what jq’s date functions expect, and then convert that to the human-readable dates you expect:
.[].published_at |= (. / 1000 | strftime("%Y-%m-%d"))
Then, we select only those elements whose dates match:
map(select(.published_at == $date))
Lastly, we put it all together, taking the $date variable from the command-line:
jq --arg date "2016-04-25" '.[].published_at |= (. / 1000 | strftime("%Y-%m-%d")) | map(select(.published_at == $date))' stuff.json