tl;dr

LocalDate.parse( 
    "January 08, 2017" , 
    DateTimeFormatter.ofPattern( "MMMM dd, uuuu" , Locale.US ) 
).format( DateTimeFormatter.BASIC_ISO_DATE ) 

Using java.time

The Question and other Answers use troublesome old date-time classes, now legacy, supplanted by the java.time classes.

You have date-only values, so use a date-only class. The LocalDate class represents a date-only value without time-of-day and without time zone.

String input = "January 08, 2017";
Locale l = Locale.US ;
DateTimeFormatter f = DateTimeFormatter.ofPattern( "MMMM dd, uuuu" , l );
LocalDate ld = LocalDate.parse( input , f );

Your desired output format is defined by the ISO 8601 standard. For a date-only value, the “expanded” format is YYYY-MM-DD such as 2017-01-08 and the “basic” format that minimizes the use of delimiters is YYYYMMDD such as 20170108.

I strongly suggest using the expanded format for readability. But if you insist on the basic format, that formatter is predefined as a constant on the DateTimeFormatter class named BASIC_ISO_DATE.

String output = ld.format( DateTimeFormatter.BASIC_ISO_DATE );

See this code run live at IdeOne.com.

ld.toString(): 2017-01-08

output: 20170108

About java.time

The java.time framework is built into Java 8 and later. These classes supplant the troublesome old legacy date-time classes such as java.util.Date, Calendar, & SimpleDateFormat.

To learn more, see the Oracle Tutorial. And search Stack Overflow for many examples and explanations. Specification is JSR 310.

The Joda-Time project, now in maintenance mode, advises migration to the java.time classes.

You may exchange java.time objects directly with your database. Use a JDBC driver compliant with JDBC 4.2 or later. No need for strings, no need for java.sql.* classes. Hibernate 5 & JPA 2.2 support java.time.

Where to obtain the java.time classes?

• Java SE 8, Java SE 9, Java SE 10, Java SE 11, and later - Part of the standard Java API with a bundled implementation.
  • Java 9 brought some minor features and fixes.
• Java SE 6 and Java SE 7
  • Most of the java.time functionality is back-ported to Java 6 & 7 in ThreeTen-Backport.
• Android
  • Later versions of Android (26+) bundle implementations of the java.time classes.
  • For earlier Android (<26), the process of API desugaring brings a subset of the java.time functionality not originally built into Android.
    • If the desugaring does not offer what you need, the ThreeTenABP project adapts ThreeTen-Backport (mentioned above) to Android. See How to use ThreeTenABP….