Recently, when looking to see if it was possible to get cURL on Android, I found something posted on June 22nd of 2013 on http://curl.haxx.se/. What was it, you may ask? It was an announcement of the curl binary version 7.31.0. This also came with a nice little surprise. cURL and openssl binaries for android! If you want, you can download them by going to http://curl.haxx.se/download.html and scrolling down until you see Linux - Android. Download the ".tar.gz" file and extract it. Inside, you will see a folder called data. To install, copy the data/local/ssl to /data/local/ssl on the device. Also, push the curl and openssl binaries from data/local/bin to /system/bin on the device. Be sure to chmod them to 0755. Once all that is done, have fun using curl!
I have confirmed this to be working on my Kindle Fire HD running android 4.2.2.
EDIT: Root is needed in order to run OpenSSL, but not cURL.
Sent from my Amazon Kindle Fire HD using xda app-developers app