Normally, when you delete a file, its name is removed from the directory listing and its disk space is made available for other files. The data remains intact on the disk and can be recovered quite easily. Confidential data, that you think has been deleted, is actually still there. This app overwrites the contents of files with random data before deleting it. You can optionally specify that files should also be overwritten with zeroes and that multiple passes of random and zero wipes be performed.
The app is open source. The source code is available at https://github.com/peterhearty/WipeFiles.
I wrote this as an exercise to learn Android programming. I don't consider it to be production quality code.
USE ENTIRELY AT YOUR OWN RISK.
The Android 4 restrictions on writing to an external SD card apply to this app.
Note that secure wiping of Solid State Disks may not always have the desired effect. When you overwrite data on an SSD, the SSD controller may move the data in order to optimise the disk's lifetime. This can result in multiple copies of the data being scattered across the disk. A wipe of free space can partially help with this but at the expense of greatly reduced disk life. Even then, SSDs contain large amounts of unaddressable redundant data that cannot be deleted by conventional means. See https://www.youtube.com/watch?v=eNlobn2CinQ for further details.