How can I export or query all hospitals from Open Street Map?

Two more ideas:

1) The read-only Overpass API. I don't know that it'll work for the whole US in one pass, but maybe if you extend the time out enough...? For a smaller area (.5 degree max) and with the benefit of a (minimal) UI, you can access Overpass via the XAPI Query Builder. There, you can put amenity=hospital in the tag search, select your area, and go.

2) Geofabrik downloads, filtered with Osmosis, as described in How to extract partial data for large regions? on the OpenStreetMap Help.

Nominatim can do this but on a large scale it would be more efficient to download the USA extracts and use your own.

Nominatim Example:

http://nominatim.openstreetmap.org/search?q=usa[hospitals]&format=json&polygon=0&addressdetails=1

Returns:

[

    {
        "place_id": "640932",
        "licence": "Data © OpenStreetMap contributors, ODbL 1.0. http://www.openstreetmap.org/copyright",
        "osm_type": "node",
        "osm_id": "257489577",
        "boundingbox": [
            "42.3371238708496",
            "42.3371276855469",
            "-71.1060409545898",
            "-71.1060333251953"
        ],
        "lat": "42.3371263",
        "lon": "-71.1060391",
        "display_name": "Children's Hospital, Blackfan Street, Fenway, Boston, Suffolk County, Massachusetts, 02215, United States of America",
        "class": "amenity",
        "type": "hospital",
        "importance": 0.44314331355821,
        "icon": "http://nominatim.openstreetmap.org/images/mapicons/health_hospital.p.20.png",
        "address": {
            "hospital": "Children's Hospital",
            "road": "Blackfan Street",
            "suburb": "Fenway",
            "city": "Boston",
            "county": "Suffolk County",
            "state": "Massachusetts",
            "postcode": "02215",
            "country": "United States of America",
            "country_code": "us"
        }

for xml replace json with xml

Source: http://wiki.openstreetmap.org/wiki/Nominatim

You might be better off asking here: https://help.openstreetmap.org/

This website can give you the correct info; http://osm2shp.ru/ it might be outdated (2016), but it will give you all the hospitals using amenity_pnt and all the ambulance stations using emergency_pnt. You can then import it into QGIS and filter it.