I’m using curl 7.43 on OS X and this works correctly (although the API response is invalid request, which is as expected) - note that Twitter does not support SSLv3 or weaker, and requires at least TLS. I suggest you look at your certificate configuration.
$ curl https://api.twitter.com/oauth/request_token -v
* Trying 104.244.42.2...
* Connected to api.twitter.com (104.244.42.2) port 443 (#0)
* TLS 1.2 connection using TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256
* Server certificate: api.twitter.com
* Server certificate: DigiCert SHA2 High Assurance Server CA
* Server certificate: DigiCert High Assurance EV Root CA
> GET /oauth/request_token HTTP/1.1
> Host: api.twitter.com
> User-Agent: curl/7.43.0
> Accept: */*
>
< HTTP/1.1 400 Bad Request
< cache-control: no-cache, no-store, must-revalidate, pre-check=0, post-check=0
< content-disposition: attachment; filename=json.json
< content-length: 62
< content-type: application/json; charset=utf-8
< date: Fri, 02 Sep 2016 12:47:15 GMT
< expires: Tue, 31 Mar 1981 05:00:00 GMT
< last-modified: Fri, 02 Sep 2016 12:47:15 GMT
< pragma: no-cache
< server: tsa_f
< set-cookie: guest_id=v1%3A147282043546846296; Domain=.twitter.com; Path=/; Expires=Sun, 02-Sep-2018 12:47:15 UTC
< status: 400 Bad Request
< strict-transport-security: max-age=631138519
< x-connection-hash: 050899943071408b0c1d2a3f6c6bccfd
< x-content-type-options: nosniff
< x-frame-options: SAMEORIGIN
< x-response-time: 100
< x-transaction: 00b4a8870065b681
< x-twitter-response-tags: BouncerCompliant
< x-xss-protection: 1; mode=block
<
* Connection #0 to host api.twitter.com left intact
{"errors":[{"code":215,"message":"Bad Authentication data."}]}%
